**5. Results and Discussions**

In this study, we investigated the unsteady magento-flow and heat transfer of Cobalt–kerosene ferrofluid past a stretchable surface. The influences of several key parameters on the dimensionless velocity *f* (<sup>τ</sup>, η), temperature <sup>θ</sup>(<sup>τ</sup>, η), skin friction *Cf*(<sup>τ</sup>, <sup>0</sup>), and Nusselt number *Nu*(<sup>τ</sup>, 0) are examined. The Lie group method is employed to reduce partial di fferential equations and local similar and non-similar models are solved employing the RK-45 technique.

The e ffects of the magnetic field *Ha* and dimensionless time τ on the velocity are symbolized in Figure 1a and on the dimensionless temperature in Figure 1b, respectively. As the time increases, the velocity at the surface rises. The magnetic field generates Lorentz strength on the fluid particles, which resist the fluid and reduce the fluid velocity, as shown in Figure 1a. Consequently, the velocity boundary layer thickness decreases. Due to the decline in velocity, the temperature increases. In the thermal boundary layer, the temperature declines to the ambient temperature. The thermal boundary layer thickness reduces with an enlargement of the dimensionless time, as exhibited in Figure 1b. The influence of the solid volume fraction of nanoparticles χ and Navierslip δ on the velocity and temperature is depicted in Figure 2a,b when τ = 0.5. In the absence of slip, the velocity is found to be higher for the pure regular fluid. At the surface, the velocity decreases with the increase in the slip and solid volume fraction, as shown in Figure 2a. No appreciable impact of χ could be observed at the surface as well as within the velocity boundary layer. The velocity boundary layer thickness enlarges with δ, which enlarges the thermal resistance and reduces the heat transfer rate; see Figure 2b. The variation of the dimensionless temperature with the solid volume fraction χ is depicted in Figure 2b. In the absence of slip, the temperature is lower at the wall and intensifies with δ. As expected, the temperature at the wall is higher for the regular fluid and dwindles with an intensify in the solid volume fraction χ. This is due to the higher thermal conductivity of Cobalt nanoparticles. With the addition of nanoparticles, the thermal conductivity of the ferrofluid increases and the heat transfer rate is enhanced.

**Figure 1.** Effects of magnetic field *Ha* and dimensionless time τ on (**a**) dimensionless velocity, and (**b**) dimensionless temperature.

**Figure 2.** Effects of solid volume fraction of nanoparticles χ and Navier slip δ on (**a**) dimensionless velocity and (**b**) dimensionless temperature.

Figure 3a,b presents the effects of radiation parameter *Rd* and Biot number *Bi* on the velocity and temperature curves. It is important to note that equations of momentum and energy are independent of each other. The momentum equation and the velocity boundary conditions are independent of the radiation parameter *Rd* and convective heating parameter *Bi*. Therefore, there is no influence of these parameters on the velocity, which is obvious from Figure 3a. On the other side, the surface temperature increases significantly with a strengthen in both *Rd* and *Bi*. As a result, the thermal boundary layer thickness is boosted, with an increase in both parameters, as depicted in Figure 3b. The radiation parameter *Rd* reveals an enhancement in radiative heat, which improves the thermal state of fluid, causing its surface temperature to increase. Similarly, as the convective heating parameter increases and tends to infinity, the convective boundary condition changes to an isothermal boundary condition.

**Figure 3.** Effects of Biot number *Bi* and radiation parameter *Rd* on (**a**) dimensionless velocity and (**b**) dimensionless temperature.

The variations in skin friction and Nusselt number with the magnetic field *Ha* are depicted in Figures 4 and 5 for different values of the velocity slip δ and the solid volume fraction χ at τ = 1 and τ = 2, respectively. In the presence of magnetic strength, a Lorentz force is generated which resists the fluid and reduces the velocity curve. Therefore, the skin friction enhances with *Ha*, as shown in Figures 4 and 5a at different dimensionless times. As expected, the skin friction increases with dimensionless time τ. In the absence of velocity slip δ, the velocity curves are higher at the surface and decline with an increment in slip parameter δ. Consequently, the skin friction declines with the boosting of slip parameter δ. For the pure regular fluid, the skin friction is lower and increases with a rise in the solid volume fraction χ. This is due to an evolution in the ferrofluid density with the increased volume fraction of cobalt nanoparticles. Figures 4 and 5b illustrate the variation of Nusselt number with the magnetic field Ha and the volume fraction of ferroparticles χ at different dimensionless times. Like skin friction, Nusselt number also increases with dimensionless time. Due to Lorentz force, the dimensionless velocity decreases and, as a result, the Nusselt number is reduced with an increasing magnetic field. Similarly, the velocity decreases due to an intension in the slip and the Nusselt number reduces. The thermal conductivity of ferroparticles increases with an increase in the volume fraction of ferroparticles. Consequently, the Nusselt number increases with increasing χ.

**Figure 4.** Effects of solid volume fraction of nanoparticles χ, magnetic Ha, and Navier slip δ parameters on (**a**) skin friction, and (**b**) Nusselt number when τ = 1.

**Figure 5.** Effects of solid volume fractions of nanoparticles χ, magnetic Ha and Navierslip δ parameters on (**a**) skin friction, and (**b**) Nusselt number when τ = 2.

Figure 6a, b presents the comparison of Nusselt numbers for kerosene oil and water for the same parameters. Due to the smaller Prandtl number *Pr* for water, the Nusselt numbers are found to be lower than kerosene. The Prandtl number *Pr* compares the rate of thermal diffusion in comparison to the rate of momentum diffusion. The higher the Prandtl number *Pr*, the higher the Nusselt number will be. It is also noticed that an increase in the Biot number Bi and radiation parameter *Rd* leads to an increase in the Nusselt number. These Nusselt numbers also become greater with increasing dimensionless time.

**Figure 6.** Effects of Biot number Bi, dimensionless time τ, and radiation parameter *Rd* on Nusselt number for (**a**) water and (**b**) kerosene oil as base fluid.
